# Full text of "Scientific Papers - Vi"

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L916]
MEMBRANES,  BARS,   AND  PLATES
431
The accompanying tables show the form of the curves of deflexion defined by (39), (40).
V	(39)	y	(39)
0°	•oooo	50	•7416
10	•1594	60	•8574
20	•3162	70	•9530
30	•4675	80	1-0217
40	•6104	90	T0518
X	(40)	X	(40)
o-o	1-0518	3-0	•1992
0-5	•9333	4-0	•0916
i-o	•7435	5-0	•0404
2-0	•4066 •	10-0	•0005
In a second communication* Mesnager returns to the question and shows by very simple reasoning that all points of a rectangular plate supported at bhe boundary move in the direction of the applied transverse forces.
If z denote V2w, then V2^, = V4'jy, is positive over the plate if the applied forces are everywhere positive. At a straight portion of the boundary of a supported plate z = 0; and this is regarded as applicable to the whole boundary Df the rectangular plate, though perhaps the corners may require further consideration. But if V2# is everywhere positive within a coutour and z vanish Dn the contour itself, z must be negative over the interior, as is physically obvious in the theory of the conduction of heat. Again, since V2w is negative throughout the interior, and w vanishes at the boundary, it follows in like manner that w is positive throughout the interior.
It does not appear that an argument on these lines can be applied to a rectangular plate whose boundary is clamped, or to a supported plate whose boundary is in part curved.
P.S. In connexion with a recent paper on the "Flow of Compressible Fluid past an Obstacle" (Phil. Mag. July 1916)I, I have become aware that :he subject had been treated with considerable generality by Prof. Cisotti of Milan, under the title " Sul Paradosso di D'Alembert" (Atti R. Istituto Veneto, 'j. Ixv. 1906). There was, however, no reference to the limitation necessary yhen the velocity exceeds that of sound in the medium. I understand that ;his matter is now engaging Prof. Oisotti's attention.
* G. R. July 24, 1916, p. 84.                  t [This volume, p. 402.]n the former w is proportional to                                                           , , ,.